Thus to estimate ETT interventional quantities \(\mathbb {E}[Y_t^{A_t=a_t^{\prime }}]\)  and \(\mathbb {E}[Y_t^{A_t=a_t^{\prime }}|a_t^{\prime }]\)  are required, at least without further assumptions. Clearly, this estimand is not a function of observed variables directly, showing that such estimation is non-identifiable from historical data collected based on the previous medication.
However if some experimental data is available, that allows us to anchor treatments to \(a_t^{\prime }\) , e.g. a randomized controlled trial, both of these quantities can be estimated, providing us with a way to counterfactually reason about the effect of alternative treatments on those who have been treated differently i.e. with \(a_t\) . In practice, some additional assumptions like monotonicity of the effects for binary treatment on the outcome can enable identifiability entirely from observational data and have been outlined in [1]}. Identifiability with a general notion of monotonicity for discrete SCMs is outlined in [2]}. These identifiability results again require ignorability/no unobserved confounding, thereby highlighting how these are closely tied to structural assumptions on the data-generating mechanisms. Further note that these identifiability results are applicable assuming treatments are binary and/or discrete. For other general estimands, it may be possible to get off-policy (i.e. from observational data) estimates of retrospective OPE estimands for specific data-generating assumptions, like no unobserved confounding and parametric assumptions like linearity.
